Simplify the following expression and state the condition under which the simplification is valid: $z = \dfrac{a^2 - a - 12}{a^2 - 4a}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{a^2 - a - 12}{a^2 - 4a} = \dfrac{(a + 3)(a - 4)}{(a)(a - 4)} $ Notice that the term $(a - 4)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(a - 4)$ gives: $z = \dfrac{a + 3}{a}$ Since we divided by $(a - 4)$, $a \neq 4$. $z = \dfrac{a + 3}{a}; \space a \neq 4$